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arXiv · 1410.4265

Independences and Partial $R$-Transforms in Bi-Free Probability

Paul Skoufranis
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Abstract

In this paper, we examine how various notions of independence in non-commutative probability theory arise in bi-free probability. We exhibit how Boolean and monotone independence occur from bi-free pairs of faces and establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove that bi-freeness is preserved under tensoring with matrices. Finally, via combinatorial arguments, we construct partial $R$-transforms in two settings relating the moments and cumulants of a left-right pair of operators.

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Paul Skoufranis. 2015-06-05. Independences and Partial $R$-Transforms in Bi-Free Probability. https://doi.org/10.1214/15-aihp691

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